The Adventures of PCA in the Decorrelation Manifold¹

A PCA decomposition of the original data can be written as

$$Y = A_P S_P$$ (2)

where the spatial maps, $S_P$ are uncorrelated and of unit variance. That is, the principle basis vectors (spatial maps) are orthonormal (i.e. orthogonal and normalised). In matrix notation this is written as $S_P S_P^T = I$.

Note that this is the same as the ICA decomposition, but uses a different function to minimise -- in this case, one that measures correlation.

The PCA decomposition is easily found using SVD. That is $Y = U D V^T$, with $U$ and $V$ being orthogonal matrices (i.e. $U U^T = V V^T = I$). Hence the PCA decomposition is given by $S_P = V^T$ and $A_P = U D$.